A337153 - OEIS

%I #12 Jan 28 2021 11:12:09

%S 1,2,25,674,32353,2426474,262059193,38522701370,7396358663041,

%T 1797315155118962,539194546535688601,195727620392454962162,

%U 84554332009540543653985,42869046328837055632570394,25206999241356188711951391673,17014724487915427380567189379274,13067308406719048228275601443282433

%N a(n) = 3^n * (n!)^2 * Sum_{k=0..n} 1 / ((-3)^k * (k!)^2).

%F Sum_{n>=0} a(n) * x^n / (n!)^2 = BesselJ(0,2*sqrt(x)) / (1 - 3*x).

%F a(0) = 1; a(n) = 3 * n^2 * a(n-1) + (-1)^n.

%t Table[3^n n!^2 Sum[1/((-3)^k k!^2), {k, 0, n}], {n, 0, 16}]

%t nmax = 16; CoefficientList[Series[BesselJ[0, 2 Sqrt[x]]/(1 - 3 x), {x, 0, nmax}], x] Range[0, nmax]!^2

%o (PARI) a(n) = 3^n * (n!)^2 * sum(k=0, n, 1 / ((-3)^k * (k!)^2)); \\ _Michel Marcus_, Jan 28 2021

%Y Cf. A000180, A073701, A336805, A337152, A337154, A337155.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Jan 27 2021